Martingale solutions to stochastic nonlocal Cahn–Hilliard- Navier–Stokes systems with singular potentials driven by multiplicative noise of jump type

Abstract

Abstract In the diffuse interface theory, the motion of two incompressible viscous fluids and the evolution of their interface are described by the well-known model H. The model consists of the Navier–Stokes equations, nonlinearly coupled with a convective Cahn–Hilliard type equation. Here we consider a stochastic version of the model driven by a noise of Lévy type, and where the standard Cahn–Hilliard equation is replaced by its nonlocal version with a singular (e.g., logarithmic) potential. The case of smooth potentials with arbitrary polynomial growth has been already analyzed in [G. Deugoué, A. Ndongmo Ngana and T. Tachim Medjo, Martingale solutions to stochastic nonlocal Cahn–Hilliard–Navier–Stokes equations with multiplicative noise of jump type, Phys. D 398 2019, 23–68]. Taking advantage of this previous result, we investigate this more challenging and physically relevant case. Global existence of a martingale solution is proved with no-slip and no-flux boundary conditions in both 2D and 3D bounded domains. In the two-dimensional case, we prove the uniqueness of weak solutions when the viscosity is constant.